Explore partial derivatives and multivariable optimization through interactive SageMath computations including critical point analysis, Lagrange multipliers, and constrained optimization problems. This hands-on Jupyter notebook covers second derivative tests, Hessian matrices, saddle point identification, and practical optimization applications in economics and engineering. CoCalc provides pre-configured computational tools for symbolic differentiation, 3D surface plotting, and gradient descent visualization, allowing students to solve complex optimization problems and understand multivariable calculus concepts through immediate computational feedback.

Published/Calculus Series/Advanced Calculus with SageMath/Advanced Calculus with SageMath - Chapter 2.ipynb

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Advanced Calculus with SageMath - Chapter 2

Multivariable Functions and Partial Derivatives

This notebook contains Chapter 2 from the main Advanced Calculus with SageMath notebook.

For the complete course, please refer to the main notebook: Advanced Calculus with SageMath.ipynb

In [3]:

# Comprehensive imports for advanced calculus
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import scipy.optimize as opt
import scipy.integrate as integrate
from scipy.integrate import solve_ivp, odeint
import sympy as sp
from sympy import *
from sage.all import *
import seaborn as sns

# Configure plotting
plt.style.use('seaborn-v0_8')
sns.set_palette("husl")
plt.rcParams['figure.figsize'] = (12, 8)

print("Advanced Calculus Environment Initialized")
print("Tools: SageMath, NumPy, SciPy, SymPy, Matplotlib")
print("Ready for multivariable calculus, vector analysis, and PDEs!")

Out[3]:

Advanced Calculus Environment Initialized
Tools: SageMath, NumPy, SciPy, SymPy, Matplotlib
Ready for multivariable calculus, vector analysis, and PDEs!

Chapter 2: Multivariable Functions and Partial Derivatives

Understanding Functions of Several Variables

A multivariable function maps points in ℝⁿ to ℝ. Examples include:

Temperature distribution: T(x,y,z,t)
Economic utility: U(x₁,x₂,...,xₙ)
Wave equations: ψ(x,y,z,t)

Partial Derivatives and the Gradient

For a function f(x,y), the gradient is: $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$

The gradient points in the direction of steepest ascent.

In [5]:

# Multivariable functions and partial derivatives
print("MULTIVARIABLE FUNCTIONS")
print("=" * 50)

# Define a multivariable function: f(x,y) = x²y + xy² - 3xy
f = x**2 * y + x * y**2 - 3*x*y
print("Function: f(x,y) =", f)

# Compute partial derivatives
df_dx = diff(f, x)
df_dy = diff(f, y)

print("\nPARTIAL DERIVATIVES")
print("∂f/∂x =", df_dx)
print("∂f/∂y =", df_dy)

# Gradient vector
gradient_f = vector([df_dx, df_dy])
print("\n∇f =", gradient_f)

# Second-order partial derivatives (Hessian matrix)
f_xx = diff(f, x, 2)
f_yy = diff(f, y, 2)
f_xy = diff(f, x, y)

hessian = matrix([[f_xx, f_xy], [f_xy, f_yy]])
print("\nHESSIAN MATRIX")
print(hessian)

# Evaluate at a specific point
point = {x: 2, y: 1}
print(f"\nAt point (2,1):")
print(f"f(2,1) = {f.subs(point)}")
print(f"∇f(2,1) = {gradient_f.subs(point)}")

Out[5]:

MULTIVARIABLE FUNCTIONS
==================================================
Function: f(x,y) = x^2*y + x*y^2 - 3*x*y

PARTIAL DERIVATIVES
∂f/∂x = 2*x*y + y^2 - 3*y
∂f/∂y = x^2 + 2*x*y - 3*x

∇f = (2*x*y + y^2 - 3*y, x^2 + 2*x*y - 3*x)

HESSIAN MATRIX
[          2*y 2*x + 2*y - 3]
[2*x + 2*y - 3           2*x]

At point (2,1):
f(2,1) = 0
∇f(2,1) = (2, 2)

In [6]:

# 3D visualization of multivariable function
def plot_3d_function():
    # Create meshgrid for plotting
    x_vals = np.linspace(-3, 3, 50)
    y_vals = np.linspace(-3, 3, 50)
    X, Y = np.meshgrid(x_vals, y_vals)
    
    # Evaluate function: f(x,y) = x^2 y + x y^2 - 3 x y
    Z = X**2 * Y + X * Y**2 - 3*X*Y
    
    # Create 3D plot
    fig = plt.figure(figsize=(15, 5))
    
    # Surface plot
    ax1 = fig.add_subplot(131, projection='3d')
    surf = ax1.plot_surface(X, Y, Z, cmap='viridis', alpha=0.8)
    ax1.set_xlabel('x')
    ax1.set_ylabel('y')
    ax1.set_zlabel('f(x,y)')
    ax1.set_title(r'Surface: $f(x,y)=x^2 y + x y^2 - 3 x y$')
    
    # Contour plot
    ax2 = fig.add_subplot(132)
    contour = ax2.contour(X, Y, Z, levels=20)
    ax2.clabel(contour, inline=True, fontsize=8)
    ax2.set_xlabel('x')
    ax2.set_ylabel('y')
    ax2.set_title('Contour Lines')
    ax2.grid(True)
    
    # Gradient field
    ax3 = fig.add_subplot(133)
    # Compute gradient at grid points
    dZ_dx = 2*X*Y + Y**2 - 3*Y
    dZ_dy = X**2 + 2*X*Y - 3*X
    
    # Subsample for clarity
    skip = 3
    ax3.quiver(X[::skip, ::skip], Y[::skip, ::skip], 
               dZ_dx[::skip, ::skip], dZ_dy[::skip, ::skip],
               alpha=0.7)
    ax3.contour(X, Y, Z, levels=10, alpha=0.3)
    ax3.set_xlabel('x')
    ax3.set_ylabel('y')
    ax3.set_title(r'Gradient Field $\nabla f$')
    ax3.grid(True)
    
    plt.tight_layout()
    plt.show()

plot_3d_function()
print("The surface shows function values, contours show level curves,")
print("and arrows show the gradient field (direction of steepest ascent)")

Out[6]:

The surface shows function values, contours show level curves,
and arrows show the gradient field (direction of steepest ascent)

Continuing Your Learning Journey

You've completed Multivariable Functions and Partial Derivatives! The concepts you've mastered here form essential building blocks for what comes next.

Ready for Optimization in Multiple Dimensions?

In Chapter 3, we'll build upon these foundations to explore even more fascinating aspects of the subject. The knowledge you've gained here will directly apply to the advanced concepts ahead.

What's Next

Chapter 3 will expand your understanding by introducing new techniques and applications that leverage everything you've learned so far.

Continue to Chapter 3: Optimization in Multiple Dimensions →

Return to Complete Course